Quantum Coherence Thermal Transistors

Shanhe Su,1 Yanchao Zhang,1 Bjarne Andresen,2 Jincan Chen1∗ 1Department of Physics and Jiujiang Research Institute, Xiamen University, Xiamen 361005, China. 2Niels Bohr Institute, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark.

(Dated: May 31, 2021) Coherent control of self-contained quantum systems offers the possibility to fabricate smallest thermal transistors. The steady coherence created by the delocalization of electronic excited states arouses nonlinear heat transports in non-equilibrium environment. Applying this result to a three- level quantum system, we show that quantum coherence gives rise to negative differential thermal resistances, making the thermal transistor suitable for thermal amplification. The results show that quantum coherence facilitates efficient thermal signal processing and can open a new field in the application of quantum thermal management devices. PACS numbers: 05.90. +m, 05.70. –a, 03.65.–w, 51.30. +i

A thermal transistor, like its electronic counterpart, is thermal flux is normally higher than the controlling (in- capable of implementing heat flux switching and mod- put) thermal flux, a thermal transistor is able to amplify ulating. The effects of negative differential thermal re- or switch a small signal. The amplification factor must sistance (NDTR) play a key role in the development of be tailored to suit specific situations. The Scovil and thermal transistors [1]. Classical dynamic descriptions Schulz-DuBois maser model is not applicable for fabricat- utilizing Frenkel-Kontorova lattices conclude that nonlin- ing thermal transistors, owing to the fact that its ampli- ear lattices are the origin of NDTR [2, 3]. Ben-Abdallah fication factor is simply a constant defined by the maser et al. introduced a distinct type of thermal transistors frequency relative to the pump frequency [18, 19]. How- based on the near-field radiative heat transfer by evanes- ever, the coherent excitation-energy transfer created by cent thermal photons between bodies [4]. Joulain et al. the delocalization of electronic excited states may aid in first proposed a quantum thermal transistor with strong the design of powerful thermal devices. Coherent con- coupling between the interacting spins, where the com- trol of a three-level system (TLS) provides us a heuristic petition between different decay channels makes the tem- approach to better understand the prime requirements perature dependence of the base flux slow enough to ob- for the occurrence of anomalous thermal conduction in tain a high amplification [5]. Zhang et al. predicted that quantum systems. asymmetric Coulomb blockade in quantum-dot thermal In this paper we design a quantum thermal transis- transistors would result in a NDTR [6]. Stochastic fluc- tor consisting of a TLS coupled to three separate baths. tuations in mesoscopic systems have been regarded as an The dynamics of the system is derived by considering the alternate resource for the fast switching of heat flows [7]. coupling between the two excited states. Steady-state Recent studies showed that quantum coherence ex- solutions will be used to prove that the coherent transi- hibits the ability to enhance the efficiency of thermal tions between the two excited states induce nonlinearity converters, such as quantum heat engines [8–10] and in nonequilibrium quantum systems. Further analysis artificial light-harvesting systems [11, 12]. Interference shows that quantum coherence gives rise to a NDTR and between multiple transitions in nonequilibrium environ- helps improve the thermal amplification. ments enables us to generate non-vanishing steady quan- Figure 1 shows the TLS modeled by the Hamiltonian tum coherence [13, 14]. Evidence is growing that long- HS as arXiv:1811.02400v1 [quant-ph] 6 Nov 2018 lived coherence boosts the transport of energy from light- X harvesting antennas to photosynthetic reaction centers HS = εi |ii hi| + ∆(|1i h2| + |2i h1|), (1) [15, 16]. The question arises whether quantum interfer- i=0,1,2 ence and coherence effects could also induce nonlinear heat conduction and enhance the performance of a ther- where ε1 (ε2) gives the energy level of the excited states mal transistor. in the molecules |1i (|2i), ε0 denotes the energy of the Scovil and Schulz-DuBois originally proposed a three- ground state |0i and is set to zero, and ∆ describes the level maser system as an example of a Carnot engine excitonic coupling between states |1i and |2i. For the and applied detailed balance ideas to obtain the maser models of biological light reactions, ∆ occurs naturally efficiency formula [17]. Because the controlled (output) as a consequence of the intermolecular forces between two proximal optical dipoles [12, 20]. In the presence of the dipole-dipole interaction, the optically excited states become coherently delocalized. |+i = cos θ |1i + sin θ |2i ∗ [email protected] and |−i = sin θ |1i − cos θ |2i are the usual eigenstates 2

and |2i h2| = sin θ sin θ |+i h+| + cos θ cos θ |−i h−| − cos θ sin θ (|+i h−| + |−i h+|) . (4) ȁퟏۧ Base Emitter The first two operators in |1i h1| and |2i h2| describe the ∆ pure dephasing of a two-level system, whereas the third ȁퟐۧ term leads to the energy exchange between the system and the base with an effective coupling proportional to ȁퟎۧ Collecter the product sin θ cos θ, i.e., X † HSB−eff = 2 sin θ cos θ (|+i h−| + |−i h+|) gk ak + ak . k (5) Figure 1. Schematic illustration of the quantum thermal transis- In reality, the TLS can be realized in the photosynthesis tor composed of a three-level system (TLS) interacting with three process. The pumping light, taking the sunlight photons baths: its ground state |0i and excited state |1i (|2i) are cou- for example, is considered the high temperature emitter. pled with the emitter (collector); the excited states |1i and |2i are The collector is formed by the surrounding electromag- diagonal-coupled with the base; and the coupling strength between |1i and |2i is characterized by ∆. netic environment which models energy transfer to the reaction center. The base provides the phonon modes coupled with the excited states. The TLS becomes irreversible due to the interaction diagonalizing the subspace spanned by |1i and |2i with with its surrounding environment. Using the Born- tan 2θ = 2∆/ (ε1 − ε2). Markov approximation, which involves the assumptions The absorption of a photon from the emitter (E) causes that the environment is time independent and the envi- an excitation transfer from the ground state |0i to the ronment correlations decay rapidly in comparison to the state |1i, whereas phonons are emitted into the base (B) typical time scale of the system evolution [22], we get the by the transitions between |1i and |2i. The cycle is closed quantum dynamics of the system in = 1 units, i.e., by the transition between |2i and |0i, and the rest of ~ the energy is released as a photon to the collector (C). dρ The Hamiltonians of the emitter, collector, and base are = −i[HS, ρ] + DE [ρ] + DB [ρ] + DC [ρ]. (6) P † † dt Hi = k ωikaikaik (i = E, C, and B), where aik (aik) refers to the creation (annihilation) operator of the bath The operators Di [ρ] (i = E, B, and C) denote the dissi- mode ωik. The TLS couples to the emitter and the col- pative Lindblad superoperators associated with the emit- lector, each constituted of harmonic oscillators, via cou- ter, base, and collector (Supplementary Eq. (S-1)), which pling constants gEk and gCk in the rotating wave approx- take the form imation, where the corresponding Hamiltonians are for- P † mally written as HSE = k gEkaEk |0i h1| + h.c. and X † 1 n † o Di [ρ] = γi (v) Ai (v) ρAi (v) − ρ, Ai (v) Ai (v) , † 2 P v HSC = k gCkaCk |0i h2| + h.c. , respectively. The (7) output of the Scovil–Schulz-DuBois maser is a radiation 0 where v = ε − ε is the energy difference between two field with a particular frequency, provided there is pop- arbitrary eigenvalues of HS, and Ai (v) is the jump op- ulation inversion between levels ε1 and ε2. In this study, erator associated with the interaction between the sys- the two excited states are coupled with a thermal reser- tem and bath i. Considering a quantum bath consisting voir, namely, the base. The interaction Hamiltonian of of harmonic oscillators, we have the decay rate γi (v) = the system with the base is described by Γ i (v) ni (v) for v < 0 and γi (v) = Γi (v) [1 + ni(v)] for v > 0 , where Γi (v) labels the decoherence rate and is X † related to the spectral density of the bath, and Ti is the HSB = (|1i h1| − |2i h2|) gBk aBk + a . (2) Bk temperature of bath i. The thermal occupation number k v/(kB Ti) in a mode is written as ni(v) = 1/ e −1 . The For a finite coupling ∆, the base modeled by Eq. (2) in- Boltzmann constant kB is set to unity in the following. duces not only decoherence but also relaxation [21]. The The steady-state populations and coherence of the counterintuitive effect of the energy exchange between open quantum system are obtained by setting the left- the two excited states and the dephasing bath becomes hand side of Eq. (6) equal zero. Then the steady state evident when the system operator coupled to the base is energy fluxes are determined by the average energy going replaced by through the TLS, i.e., . X |1i h1| = cos θ cos θ |+i h+| + sin θ sin θ |−i h−| E(∞) = Tr{HSDi [ρ (∞)]} = JE + JC +JB = 0 i=E,C,B + sin θ cos θ (|+i h−| + |−i h+|) (3) (8) 3 which complies with the 1st law of thermodynamics. The the base temperature TB. Then the amplification factor heat fluxes JE , JC , and JB are defined with respect to αE/C explicitly reads their own dissipative operators. Thus,

∂JE/C αE/C = . (12) nE ∂JB JE = −ΓE (ε1)(nE + 1) ε1 ρ1 − ρ0 + ∆< (ρ12) nE + 1 Comparison of the slopes of the thermal currents is the = JE1 + JE2, (9) key parameter to ﬁnd out whether the ampliﬁcation ef-

fect exists. When αE/C > 1, a small change in JB stimulates a large variation in JE or JC and the ther- n mal transistor effect appears. This implies that a small J = −Γ (ε )(n + 1) ε ρ − C ρ + ∆< (ρ ) C C 2 C 2 2 n + 1 0 12 change of the heat flux signal of the base would lead C to noticeable changes of the energy flowing through the = JC1 + JC2, (10) emitter and collector. We consider heat fluxes from the baths into the TLS and as positive. As TE and TC are fixed values and TB is ad- justable, the thermal conductances of the three terminals ε − ε are defined as J = −Γ (ω) sin2 2θ(2n + 1)[ 1 2 (ρ − ρ ) B B B 2 11 22 q 2 2 ∂Ji (ε1 − ε2) /4 + ∆ σi = − = σi1 + σi2, (13) + + 2∆< (ρ12)] = JB1 + JB2. ∂TB 2nB + 1 (11) ∂Jij σij = − (i = E,C,B; j = 1, 2) σi1 where ∂TB , are the thermal conductances with respect to the spontaneous The three heat fluxes are no longer linear functions of emission, and σi2 are the thermal conductances relying the rate of the spontaneous emission, indicating that the on the coherence < (ρ12). Using Eq. (13), the amplifica- symmetric property is closely related to the base induced tion factor in Eq. (12) can be recast in terms of σE and coherence of the excited states. In Eqs. (9) − (11), σC , i.e., each heat flux is divided into two categories. The terms Ji2 (i = E,C,B) are connected to the coherence in the σE/C < (ρ ) ρ J αE/C = − . (14) local basis, i.e., 12 (the real part of 12). i1 is the σC + σE remainder components depending on the populations of the TLS. The absolute value of the amplification factor αE/C > 1 The thermodynamics of a TLS was originally proposed implies that one of the thermal conductances is negative, by Scovil and Schulz-DuBois [17]. Boukobza et al. ob- i.e., σC < 0 or σE < 0. This means that there exists tained the Scovil–Schulz-DuBois maser efficiency formula a NDTR, and consequently, the TLS can behave as a when the TLS was operated as an amplifier [18, 23, 24]. thermal transistor by controlling the heat flow in analogy The efficiency of the amplifier is defined as the ratio of the to the usual electric transistor. output energy to the energy extracted from the hot reser- In the following section, we need to explore the extent voir [25]. In a nonequilibrium steady state, the efficiency to which the quantum nature of the TLS affects the ther- is a fixed value which equals 1 − (ε2 − ε0) / (ε1 − ε0), be- mal transistor. The formalism obtained here will allow cause all heat fluxes are linear functions of the same rate us to access how coherences can lead to a NDTR and of excitation. However, a thermal transistor is a thermal an enhancement of the amplification factor. To do so, device used to amplify or switch the thermal currents at the thermal conductances and temperatures of the three the collector and the emitter via a small change in the baths are recast in units of ∆. In the wide-band ap- base heat flux or the base temperature. Nonlinearity is proximation, we write the decoherence rates of the three the essential element needed to give rise to such ther- terminals as Γi (v) = Γi and the dephasing rate of the mal amplification. For the purpose of flexible control of base as γB (0) = γ0 . the thermal currents, the characteristic functions of the Figure 2(a) shows the thermal conductances σi of each TLS should not entirely depend upon the energy level terminal as functions of the base temperature TB. |σE|, structure of the TLS. σC , and σB decrease with TB at low temperature and A thermal amplifier requires a transistor with a high become constant as TB approaches TE. As expected, σB amplification factor αE/C , which is defined as the instan- remains lower than |σE| and σC over the whole range. taneous rate of change of the emitter or collector heat flux A tiny change of the base heat flux JB or temperature to the heat flux applied at the base. The quantum ther- TB is able to dramatically change the emitter and col- mal transistor has fixed emitter and collector tempera- lector thermal flows JE and JC , leading to a noticeable tures TE and TC (TE > TC ), respectively. The fluxes JE amplification effect. Similar to the decomposition of the and JC are controlled by JB, which can be adjusted by thermal fluxes, each thermal conductance can be divided 4

풂 × ퟏퟎ−ퟑ 풃 × ퟏퟎ−ퟑ

−ퟑ 풄 × ퟏퟎ−ퟑ 풅 × ퟏퟎ

Figure 3. The amplification factors αE (solid line) and αC (dashed line) versus the base temperature TB . All parameters are the same as those used in Fig. 2 Figure 2. (a) The overall thermal conductances σi; (b) the thermal conductances σi1; (c) the thermal conductances σi2; and (d) 풂 풄 the real part of the coherence < [ρ12] versus the base temperature × ퟏퟎ−ퟑ TB . We choose the parameters in units of ∆: ΓE /∆ = Γ C /∆ = ΓB /∆ = γ0/∆ ≡ 1, ε1/∆ = 10, ε2/∆ = 7, TE = ∆/0.003, and TC = ∆/0.15. 풃 into two separate parts. Figures 2 (b) and (c) display the thermal conductances σi1 pertaining to the population distributions and to the coherence contributed thermal Figure 4. (a) The overall thermal conductances σ ; (b) the thermal conductances σi2 varying with the base temperature TB. i conductances of the base σBj (inset); and (c) the amplification σE1, σC1, and σB1 share a magnitude close to each other, factors α (solid line) and α (dashed line) versus the decoherence indicating that it is unlikely to create an autonomous E C rate of the base ΓB . thermal amplifier without coherence. Quantum coherence < (ρ12) exists [Fig. 2(d)], allowing us to modify the thermodynamic behavior through the quantum control. in the large-ΓB regime (ΓB > 1.287Δ), while αC tends For the two thermal conductances σB1 and σB2 of the to divergence for ΓB → 1.287Δ. The amplification factor base, σB1>0 [Fig. 2 (b)], whereas σB2 originating from αE as a function of ΓB has opposite signs. The deco- the coherence is negative [Fig. 2(c)], ensuring that we herence rate ΓB is an important parameter for building σ achieve a vanishing B [Fig. 2(a)]. Such a phenomenon a desirable amplifier. As illustrated in Figure 4(b), the makes large thermal amplifications possible. thermal conductance σB of the base is the sum of σB1 and The curves of the amplification factors αE and αC as σB2. Once again, we observe that σB1 is always positive, functions of the base temperature TB are illustrated in while the thermal conductance relevant to the coherence Fig. 3. The amplification factors αE and αC are clearly effect σB2<0 leading to a cancellation of the sum when greater than 1 over a large range of TB. As seen from Eq. ΓB → 1.287Δ. For the same reason, the amplification (14), these effects result from σE < 0, which is similar to factors diverge at ΓB → 1.287Δ when σB = 0. the property of some electrical circuits and devices where Coherence is maintained in a nonequilibrium steady an increase in voltage across the overall assembly results state even in the presence of the dephasing bath. How- in a decline in electric current through it, i.e., negative ever, a large dephasing rate has a deleterious effect on the differential conductance. Specifically, Fig. 3 shows that characteristics of the TLS thermal transistor [Fig. 5(b)]. the amplification factors diverge at TB = 135.3∆ due to Figure 5(a) shows that the absolute value |ρ12| and the the fact that the thermal conductance of the base σB = 0, real part < [ρ12] of coherence are monotonically decreas- induced by the quantum coherence. Under these condi- ing functions of γ0, the decoherence rate of the base. The tions, an infinitesimal change in JB makes a considerable pure-dephasing bath acting on the TLS induces the loss J J difference in E and C . of steady coherence, yielding smaller αE/C . Figures 4 and 5 reveal the influences of the decoherence In summary, we build a TLS to analyze the effects of rate ΓB and the dephasing rate of the base γ0 on the per- the dipole–dipole interaction and the dephasing on the formance of the thermal transistor. The base tempera- energy transfer processes in a thermal transistor. The ture TB = ∆/0.015, while the values of other parameters coupling between the two excited states of the TLS is ca- are the same as those used in Fig. 2. The amplifica- pable of generating steady coherence in a nonequilibrium tion factor αC increases as ΓB increases in the small-ΓB environment, making the thermal fluxes behave nonlin- regime (ΓB < 1.287Δ), but it decreases as ΓB increases early. The coherence, at the same time, gives rise to 5

풂 풃 NTDR of the base. Quantum coherence enables the thermal flow through the collector and emitter to be controlled by a small change in the heat flux through the base. Such a thermal transistor can amplify a small in- put signal as well as direct heat to flow preferentially in one direction. The thermal transistor effect can be sig- nificantly improved by optimizing the base temperature and coherence rate or reducing the dephasing rate.

Figure 5. (a) The absolute value and the real part of coherence, |ρ | and < [ρ ], versus the dephasing rate of the base γ . (b) The We thank Dr. Dazhi Xu for helpful discussions. This work 12 12 0 has been supported by the National Natural Science Foundation of ampliﬁcation factors α (solid line) and α (dashed line) versus E C China (Grant No. 11805159) and the Fundamental Research Fund the dephasing rate of the base γ0. for the Central Universities (No. 20720180011).

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